On the kernel of the reciprocity map of normal surfaces over finite fields (Q1818745)

Let \(X\) be a proper variety over a finite field. Let \(\operatorname {CH}_0(X)\) denote the Chow group of zero cycles on \(X\), and let \(\pi_1^{\text{ab}}(X)\) be the abelian étale fundamental group of \(X\). Let \(\Theta_X: \operatorname {CH}_0(X)\to \pi_1^{\text{ab}}(X)\) be the reciprocity map. This map has dense image if \(X\) is normal [\textit{S. Lang}, Ann. Math., II. Ser. 64, 285-325 (1956; Zbl 0089.26201)], and is injective if \(X\) is smooth [\textit{K. Kato} and \textit{S. Saito}, Ann. Math., II. Ser. 118, 241-275 (1983; Zbl 0562.14011)]. The authors investigate the kernel of \(\Theta_X\) if \(X\) is a normal surface (theorem 0.1). Key ingredients in the proof are: the existence of desingularization of two-dimensional schemes (due to Lipman) and a cohomological Hasse principle for a smooth proper surface (due to Colliot-Thélène-Sansuc-Soulé/Kato). Fix a resolution \(\rho: S\to X\), and let \(D= \rho^{-1} (x_0)_{\text{red}}\) be the exceptional fiber (\(x_0\in X\) denotes an isolated singular point). Let \(H_1 (\Gamma_D,\mathbb{Z})\) denote the first homology group of the dual graph of \(D\). The natural homomorphism \(\partial_X: H_1 (\Gamma_D,\mathbb{Z})\to \operatorname {CH}_0(X)\) (constructed in section 2.2) plays a key role in the study of \(\operatorname {Ker} (\Theta_X)\). Two types of normal projective surfaces are constructed whose reciprocity maps are not injective (section 4).